Make the rosemary simple syrup: In a medium saucepan set over medium heat, combine the rosemary, sugar and 2 cups water and bring to a simmer. Simmer for 20 minutes, then remove the pan from the heat and set aside to cool to room temperature. Strain the syrup through a fine-mesh sieve and store in a covered container or bottle. The rosemary syrup will keep, refrigerated, up to 2 weeks.
Rinse a Collins glass with the Chartreuse and discard the excess. In a cocktail shaker filled with ice, combine the sotol, lime juice, lemon juice, ½ ounce of the rosemary syrup and the egg white. Cover and shake for 2 minutes. Strain into the Chartreuse-rinsed glass. Add a dash of the bitters on top and swirl delicately. Garnish with the rosemary sprig and serve.